Is it possible to combine the F-test and t-test to test for differences in both the variance and the mean? I would like to test $H_{0} :$ $\mu_{1} = \mu_{2}, \ \sigma_{1}^{2} = \sigma_{2}^{2}$ vs $H_{0} :$ $\mu_{1} \neq \mu_{2}, \ \sigma_{1}^{2} \neq \sigma_{2}^{2}$Is there a test statistic that would allow me to detect both differences in the mean and differences in the variance? Is there a way that I could possibly combine the $F$ - $test$ and the $t$ - $test$ to derive this test statistic? I am assuming that the population is normally distributed but what if it wasn't?I am very stuck so any help would be greatly appreciated.
 A: This problem of detecting a difference in two Normal distributions based on independent random samples from each was solved asymptotically by Pearson and Neyman in 1930 using the likelihood ratio test. (Specifically, the alternative hypothesis is that $\mu_1\ne\mu_2$ or $\sigma_1^2\ne\sigma_2^2.$  Under the null hypothesis, the likelihood ratio statistic is asymptotically uniform as the sizes of both samples grow large.) In 2012, Zhang, Xu, and Chen provided a tractable expression for the CDF of the likelihood ratio statistic: it is a double integral that they compute numerically.
For other parametric families of distributions the problem can, in principle, be solved with similar techniques.

Let the two samples be $(x_i|1\le i\le n)$ from a Normal$(\mu_1, \sigma_1^2)$ distribution and $(y_j|1\le j\le m)$ from a Normal$(\mu_2, \sigma_2^2)$ distribution.  The likelihood ratio statistic is
$$\lambda_{n,m} = \frac{s_x^{n/2} s_y^{m/2}}{s^{(n+m)/2}}$$
where
$$s_x = \sum_{i=1}^n(x_i-\bar{x})^2/n,$$
$$s_y = \sum_{j=1}^m(y_j - \bar{y})^2/m,$$
$$s = \left(\sum_{i=1}^n(x_i-u)^2 + \sum_{j=1}^m(y_j-u)^2\right)/(n+m)$$
and $\bar{x}, \bar{y},$ and $u$ are the sample means of the $x_i,$ the $y_j,$ and the combined sample, respectively.  The CDF of $\lambda_{n,m}$ is
$$F_{n,m}(\lambda) = 1 - C\iint_D\frac{w_1^{(n-1)/2}w_2^{(m-1)/2}}{\sqrt{1-w_1-w_2}}\frac{dw_1}{w_1}\frac{dw_2}{w_2}$$
with
$$C = \frac{\Gamma(\frac{n+m-1}{2})}{\Gamma(\frac{n-1}{2})\Gamma(\frac{m-1}{2})\Gamma(\frac{1}{2})},$$
$$D = \left\{(w_1,w_2)|w_1\gt 0, w_2\gt 0, w_1+w_2\lt 1, \frac{(n+m)^{(n+m)/2}}{n^{n/2}m^{m/2}}w_1^{n/2}w_2^{m/2}\gt \lambda\right\}.$$

Plot of the CDF of the likelihood ratio statistic $\lambda_{2,2}$.  The curve is calculated using numeric integration at the points $\lambda=0, 1/20, 2/20, \ldots, 1$ while the points are the result of a simulation of $10,000$ pairs of samples.

References
Lingyun Zhang, Xinzhong Xu, and Gemai Chen.  The Exact Likelihood Ratio Test for Equality of Two Normal Populations.  The American Statistician, August 2012, Vol. 66, No.3 pp 180-184.
E. S. Pearson and J. Neyman, On the Problem of Two Samples (1930).  Published in Joint Statistical Papers (1967), eds. J. Neyman and E. S. Pearson, Cambridge University Press, pp 99-115.
